To prove statement (U3), take \(N\)  to be sufficiently large and construct a Rohlin tower \(\lbrace C, TC,\dots , T^{2N-1}C\rbrace \)  for \((X,\mu ,T)\)  where the \(T^{i}C\)  are mutually disjoint clopen sets such that \(\mu (X \setminus \bigcup _{i=0}^{2N-1}T^{i} C) > 1-\varepsilon \) . Let \(D=\bigcup _{i=0}^{N-1}T^{2i}C\) . Then \(\mu (C)\ge \frac{1-\varepsilon }{2N}\) , \(\mu (D) \ge N\cdot \mu (C) \ge N \frac{1-\varepsilon }{2N} = \frac{1-\varepsilon }{2}\) , and \(D\cap TD = \varnothing \) . Such a Rohlin tower may be constructed by following the proof of [1]}, starting with a clopen set \(A\)  in the proof.
